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Full text of "Treatise On Analysis Vol-Ii"

70       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

(d)   Let E, F be two topological vector spaces and/:E-F a continuous linear
mapping. Then for each bounded subset B of E, the image set/(B) is bounded in F.

7, (a) Let E be a (real or complex) Hausdorff topological vector space. Show that, if
there exists a bounded neighborhood V of 0 in E (Problem 6), the sets w~1V (n an
integer >0) form a fundamental system of neighborhoods of 0 in E, and consequently
E is metrizable. If furthermore E is locally convex, then its topology can be defined
by a single norm.

(b)    Let E be a metrizable locally convex space whose topology is defined by an
increasing sequence (pn) of seminorms. Show that the topology of E can be defined
by a single norm if and only if there exists an integer N such that, for all n ^> N, there
exists a real number kn ^ 0 for which pn <ji knp^.

(c)    Let E be the real vector space of indefinitely differentiable real-valued functions
on the interval I  [0,1 ] in R. For each n J> 0 and each/e E, put

/>(/)= sup (sup|D*/(r)|)

04*<nVreI             /

(where D/=/). Show that each pn is a norm on E and that the topology defined by
the sequence of norms (pn) cannot be defined by a single norm (cf. (17.1)).
(d) Let E be a Hausdorff locally convex space whose topology is defined by a se-
quence of seminorms but cannot be defined by a single norm. If d is a translation-
invariant distance defining the topology of E (12.9.2) show that every set in E which
is bounded in the sense of the topological vector space structure of E is bounded with
respect to the distance d, but there exist subsets of E which are bounded with respect
to dbut are unbounded with respect to the topological vector space structure of E.

8.    Let E be a metrizable topological vector space.

(a)    Show that every balanced subset of E which absorbs all sequences in E which
converge to 0 is a neighborhood of 0 in E. Deduce that, if # is a linear mapping of
E into a topological vector space F, and w transforms every sequence converging to 0
in E into a bounded sequence in F, then w is continuous.

(b)    Let (Bn) be any sequence of bounded subsets of E. Show that there exists a
sequence (An) of nonzero scalars such that the union of the sets AnBn is bounded.

9.    Let E be a topological vector space. A set S3 of bounded subsets of E is called a, funda-
mental system of bounded subsets if every bounded set in E is contained in a set
belonging to 83. Show that, if E is a metrizable locally convex space whose topology
cannot be defined by a single norm, no fundamental system of bounded subsets of E
is denumerable (use Problem 8).

10,   Let (EM)n2ii be a sequence of normed spaces such that Ex is not separable. Let F be a
vector subspace of E =O E > such that PrB(F) = En for each n. Show that there exists

n

a real number 8 > 0 and a bounded sequence (xm)mzi in F such that

whenever / ^j. (Remark that, for each index n and eachnondenumerable subset A of
F, there exists a nondenumerable subset B of A such that prn(B) is bounded in En .)
Deduce that a Frechet space is separable if every bounded subset is relatively
compact.

11.   A subset A of a real or complex vector space E is convex if, whenever x and y belong
to A and 0 <; A <; 1, we have AJC + (1  X)y e A (Section 8.5, Problem 9). It followsounded if and only if pra(B) is bounded